Our concern is for income inequalities that may result from non-welfaristic redistribution schemes. We show that for large classes of income functions Lorenz dominance results can be found in the comparison of two egalitarian equivalent mechanisms. Comparisons of different conditionally egalitarian mechanisms only yield poverty dominance results. In general, no egalitarian equivalent mechanism can be Lorenz dominated by a conditionally egalitarian mechanism. Our analysis stresses the need for accurate empirical estimates of the pre-tax income function and of the distributions of responsibility and compensation characteristics.